\u56fe $G$ \u7531\u9876\u70b9\u96c6 $V$ \u548c\u8fb9\u96c6 $E$ \u6784\u6210，\u8bb0\u4e3a $G=(V,E)$\u3002<\/p>\n
\u5b58\u50a8\u7ed3\u6784：<\/p>\n
const int MAXN = 1000;
\nint graph[MAXN][MAXN]; \/\/ \u6743\u91cd\u5b58\u4e8e\u77e9\u9635<\/p>\n
#include <vector>
\nusing namespace std;<\/p>\n
struct Edge {
\n int to;
\n int weight;
\n};<\/p>\n
vector<vector<Edge>> adjList; \/\/ adjList[i]\u5b58\u50a8\u9876\u70b9i\u7684\u90bb\u63a5\u8fb9<\/p>\n
vector<bool> visited;<\/p>\n
void dfs(int u) {
\n visited[u] = true;
\n for (auto& edge : adjList[u]) {
\n if (!visited[edge.to]) {
\n dfs(edge.to);
\n }
\n }
\n}<\/p>\n
https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MDQ4MS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MDcwMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1Mjk4MS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1Mjk4My5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1Mjk4NS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIyMS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIyNC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzMxOC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIzMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzE3Ni5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzA3Ny5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIzOS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI0NC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI1MC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzE5Ny5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzM0Ni5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI1OS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUwNS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI2OC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUxMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzQxMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzM2Ni5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzQyMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUzMC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUzMS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI5My5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzQzNi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzYwMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkyMzY0LzY5NTk1MzAwMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDM4OTIwMjE2LzY5NTk1MjkzOS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkyMzY0LzY5NTk1MDk0Ny5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQxNTQ4MjkwLzY5NTk1MTAyMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkyMzY0LzY5NTk1Mjc0OC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQxNTQ4MjkwLzY5NTk1MjY2Mi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQxNTQ4MjkwLzY5NTk1MjkyOS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDM4OTIwMjE2LzY5NTk1MjY2MS5zaHRtbA==.html \u00a0<\/p>\n
\u5e94\u7528\u573a\u666f：\u8fde\u901a\u5206\u91cf\u68c0\u6d4b\u3001\u62d3\u6251\u6392\u5e8f\u3001\u56de\u6eaf\u6cd5<\/p>\n
#include <queue>
\nvoid bfs(int start) {
\n queue<int> q;
\n q.push(start);
\n visited[start] = true;<\/p>\n
while (!q.empty()) {
\n int u = q.front();
\n q.pop();
\n for (auto& edge : adjList[u]) {
\n if (!visited[edge.to]) {
\n visited[edge.to] = true;
\n q.push(edge.to);
\n }
\n }
\n }
\n}<\/p>\n
\u5e94\u7528\u573a\u666f：\u6700\u77ed\u8def\u5f84（\u65e0\u6743\u56fe）\u3001\u5c42\u7ea7\u904d\u5386<\/p>\n
\u6838\u5fc3\u601d\u60f3：\u8d2a\u5fc3\u7b56\u7565，\u9002\u7528\u4e8e\u975e\u8d1f\u6743\u56fe $$d[v] = \\\\min(d[u] + w(u,v))$$<\/p>\n
#include <queue>
\n#include <climits><\/p>\n
vector<int> dijkstra(int src, int n) {
\n vector<int> dist(n, INT_MAX);
\n dist[src] = 0;
\n priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq;
\n pq.push({0, src});<\/p>\n
while (!pq.empty()) {
\n auto [d, u] = pq.top();
\n pq.pop();
\n if (d != dist[u]) continue; \/\/ \u65e7\u6570\u636e\u8df3\u8fc7
\n for (auto& edge : adjList[u]) {
\n int newDist = d + edge.weight;
\n if (newDist < dist[edge.to]) {
\n dist[edge.to] = newDist;
\n pq.push({newDist, edge.to});
\n }
\n }
\n }
\n return dist;
\n}<\/p>\n
\u9002\u7528\u6761\u4ef6：\u6709\u5411\u65e0\u73af\u56fe（DAG） $$ \\\\text{\u5165\u5ea6} \\\\operatorname{deg}^{-}(v) = 0 \\\\implies \\\\text{\u53ef\u79fb\u9664} $$<\/p>\n
vector<int> topologicalSort(int n) {
\n vector<int> indegree(n, 0);
\n vector<int> result;
\n queue<int> q;<\/p>\n
\/\/ \u8ba1\u7b97\u5165\u5ea6
\n for (int u = 0; u < n; u++)
\n for (auto& edge : adjList[u])
\n indegree[edge.to]++;<\/p>\n
\/\/ \u5165\u5ea6\u4e3a0\u5165\u961f
\n for (int i = 0; i < n; i++)
\n if (indegree[i] == 0) q.push(i);<\/p>\n
while (!q.empty()) {
\n int u = q.front();
\n q.pop();
\n result.push_back(u);
\n for (auto& edge : adjList[u]) {
\n if (–indegree[edge.to] == 0)
\n q.push(edge.to);
\n }
\n }
\n return result.size() == n ? result : vector<int>{}; \/\/ \u5224\u65ad\u73af
\n}<\/p>\n
https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MDQ4MS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MDcwMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1Mjk4MS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1Mjk4My5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1Mjk4NS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIyMS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIyNC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzMxOC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIzMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzE3Ni5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzA3Ny5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzIzOS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI0NC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI1MC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzE5Ny5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzM0Ni5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI1OS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUwNS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI2OC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUxMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzQxMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzM2Ni5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzQyMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUzMC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzUzMS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzI5My5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzQzNi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkzMDExLzY5NTk1MzYwMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkyMzY0LzY5NTk1MzAwMy5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDM4OTIwMjE2LzY5NTk1MjkzOS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkyMzY0LzY5NTk1MDk0Ny5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQxNTQ4MjkwLzY5NTk1MTAyMi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQyMDkyMzY0LzY5NTk1Mjc0OC5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQxNTQ4MjkwLzY5NTk1MjY2Mi5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDQxNTQ4MjkwLzY5NTk1MjkyOS5zaHRtbA==.html https:\/\/tv.sohu.com\/v\/dXMvNDM4OTIwMjE2LzY5NTk1MjY2MS5zaHRtbA==.html \u00a0<\/p>\n
\u76ee\u6807：\u8fde\u63a5\u6240\u6709\u9876\u70b9\u7684\u6700\u5c0f\u6743\u91cd\u8fb9\u96c6 $$ T \\\\subseteq E, \\\\quad \\\\min \\\\sum_{e \\\\in T} w(e) $$<\/p>\n
int prim(int n) {
\n vector<bool> visited(n, false);
\n vector<int> minDist(n, INT_MAX);
\n minDist[0] = 0;
\n int total = 0;<\/p>\n
for (int i = 0; i < n; i++) {
\n int u = -1;
\n for (int j = 0; j < n; j++) \/\/ \u9009\u6700\u5c0f\u672a\u8bbf\u95ee\u70b9
\n if (!visited[j] && (u == -1 || minDist[j] < minDist[u]))
\n u = j;<\/p>\n
if (minDist[u] == INT_MAX) return -1; \/\/ \u4e0d\u8fde\u901a
\n visited[u] = true;
\n total += minDist[u];
\n for (auto& edge : adjList[u])
\n if (edge.weight < minDist[edge.to])
\n minDist[edge.to] = edge.weight;
\n }
\n return total;
\n}<\/p>\n
\u8def\u5f84\u89c4\u5212<\/p>\n
\u4f9d\u8d56\u5206\u6790<\/p>\n
\u7f51\u7edc\u4f18\u5316<\/p>\n
\u63d0\u793a：\u5b9e\u9645\u7f16\u7801\u9700\u5904\u7406\u8fb9\u754c\u6761\u4ef6（\u5982\u4e0d\u8fde\u901a\u56fe\u3001\u81ea\u73af\u8fb9），\u5efa\u8bae\u4f7f\u7528C++ STL\u7684priority_queue\u4f18\u5316Dijkstra，\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 $O(|E|\\\\log|V|)$\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u56fe\u8bba\u57fa\u7840\u7406\u8bba\u4e0e\u5b9e\u9645\u5e94\u7528\u8be6\u89e3（C\u5b9e\u73b0）
\n\u4e00\u3001\u57fa\u672c\u6982\u5ff5
\n\u56fe $G$ \u7531\u9876\u70b9\u96c6 $V$ \u548c\u8fb9\u96c6 $E$ \u6784\u6210，\u8bb0\u4e3a $G(V,E)$\u3002
\n\u6709\u5411\u56fe：\u8fb9\u6709\u65b9\u5411，$E \\\\subseteq V \\\\times V$\u65e0\u5411\u56fe：\u8fb9\u65e0\u65b9\u5411，$E$ \u4e3a\u65e0\u5e8f\u5bf9\u96c6\u5408\u6743\u91cd\u56fe：\u8fb9\u9644\u52a0\u6743\u503c\u51fd\u6570 $w: E \\\\to \\\\mathbb{R}$
\n\u5b58\u50a8\u7ed3\u6784：
\n\u90bb\u63a5\u77e9\u9635（\u7a20\u5bc6\u56fe<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55,1984,427],"topic":[],"class_list":["post-64234","post","type-post","status-publish","format-standard","hentry","category-server","tag-c","tag-1984","tag-427"],"yoast_head":"\n